Problem: Assume that $S$ is an outwardly oriented, piecewise-smooth surface with a piecewise-smooth boundary curve $C$ oriented negatively with respect to the orientation of $S$. Let $F$ be a vector field in $\mathbb{R}^3$. Does Stokes' theorem necessarily apply to the surface $S$, boundary curve $C$, and vector field $F$ ? Choose 1 answer: Choose 1 answer: (Choice A) A Yes (Choice B) B No
Explanation: Assume we have a continuously differentiable three-dimensional vector field $F(x, y, z)$, an oriented piecewise-smooth surface $S$, and a piecewise-smooth, simple, closed boundary curve $C$ oriented positively with respect to $S$. Then Stokes' theorem states that we have the equality below: $ \oint_C F \cdot dr = \iint_S \text{curl}(F) \cdot dS$ If $C$ is negatively oriented, the line integral is equal to the negative of the double integral. [What does any of that mean?] The problem never says that $C$ is simple or closed, and it also never specifies that $F$ is continuously differentiable. Because we can't guarantee that Stokes' theorem is valid in this case, the answer is no, we can't necessarily apply it.